This publication makes a speciality of mathematical difficulties relating various purposes in physics, engineering, chemistry and biology. It covers subject matters starting from interacting particle platforms to partial differential equations (PDEs), statistical mechanics and dynamical platforms. the aim of the second one assembly on Particle structures and PDEs used to be to collect popular researchers operating actively within the respective fields, to debate their subject matters of craftsmanship and to provide fresh clinical leads to either components.

We use characters of lattices (i. e. lattice morphisms into
the point lattice 2) and characters of topological areas
(i. e. non-stop features into an thoroughly topologized
element area 2) to acquire connections and dualities among
various different types of lattices and topological areas. The
objective is to provide a unified therapy of varied identified
aspects within the relation among lattices and topological areas
and to find, at the approach, a few new ones.

It follows that they are not isomorphic (to see this one can notice that Gr(1,1,1,2) (E) is Fano, while Gr(1,1,1,2) (F ) is not). 1. Positivity. In this section we prove that quiver Grassmannians which are smooth of minimal dimension have positive Euler characteristic. This is based on the following key result. 5. For every indecomposable representation M of a Dynkin quiver Q, and every dimension vector e, the quiver Grassmannian Gre (M ) has zero odd cohomology. In particular, χ(Gre (M )) ≥ 0.

In other words, the map Hom(N, π) : HomQ (N, E) → HomQ (N, M ) induced by π is surjective and its kernel is HomQ (N, τ M ). From this we see that [N, τ M ⊕ M ] = [N, E]. 1 this yields an embedding of N into E, contradicting the emptiness of GrdimM (E). Thus N M . Since [M, τ M ] = 0, the only embedding of M into τ M ⊕ M is the canonical one, proving that GrdimM (τ M ⊕ M ) is just a point. The tangent space at this point is isomorphic to HomQ (M, τ M ) which is zero dimensional, proving that GrdimM (τ M ⊕ M ) is a reduced point.